Fixed point theorem of Leggett-Williams type and its application
One of the generalizations of Krasnoselskii's theorem on cone expansion and compression was obtained in [\textit{R. W. Leggett, L. R. Williams}, J. Math. Anal. Appl., Vol. 76, 91--97 (1980; Zbl 0448.47044)]. In the present paper, the author proves the following Leggett-Williams type theorem: Theorem. Let \(P\) be a normal cone in a Banach space \(E\) and let \(\gamma\) be the normal constant of \(P\). Assume that \(\Omega_1\) and \(\Omega_2\) are bounded open sets in \(E\) such that \(0\in \Omega_1\) and \(\overline{\Omega}_1\subset \Omega_2\). Let \(F:P\cap(\overline{\Omega}_2\backslash\Omega_1)\to P\) be a completely continuous operator and \(u_0\in P\backslash {0}\). If either \( \gamma\| x\| \leq \| Fx\| \) for \(x\in P(u_0)\cap\partial\Omega_1\) and \(\| Fx\| \leq\| x\| \) for \(x\in P\cap\partial\Omega_2\), or \(\| Fx\| \leq\| x\| \) for \(x\in P\cap\partial\Omega_1\) and \(\gamma\| x\| \leq \| Fx\| \) for \(x\in P(u_0)\cap\partial\Omega_2\) is satisfied, then \(F\) has a fixed point in the set \(P\cap (\overline{\Omega}_2 \backslash \Omega_1)\). The author applies this theorem to prove the existence of positive solutions to the following second order three-point boundary value problem: \[ x^{\prime\prime}(t)+f(t, x(t))=0, \] \[ x(0)=0, \;\;\alpha x(\eta)=x(1), \] where \(t\in [0,1]\), \(f\) is a continuous function, \(\eta\in (0,1)\), \(\alpha\geq 0\), and \(1-\alpha\eta>0\).
scientific article; zbMATH DE number 2141782